Reversible.jump Marko' chain Monte Carlo (RJMCMC) methods are used to fit Bayesian capture recapture models incorporating heterogeneity in individuals and samples. Heterogeneity in capture probabilities comes from finite mixtures and/or fixed sample effects allowing for interactions. Estimation by RJMCMC allows automatic model selection and/or model averaging. Priors on the parameters stabilize the estimates and produce realistic credible intervals for population size for overparameterized models; in contrast to likelihood-based methods. To demonstrate the approach we analyze the standard Snowshoe hare and Cottontail rabbit data sets from ecology, a reliability testing data set.

For repairable products, the warrantor has options in choosing the type of repair performed to an item that fails within the warranty period. The focus is on a particular warranty repair strategy, related to the degree of the warranty repair, under a non-renewing two-dimensional warranty policy that is free of charge to the consumer. A rectangular warranty region, as in the automotive industry, is considered and partitioned into disjoint subregions. Each of these subregions has a preassigned degree of repair for a faulty item. First, for a partition of size n, an expression is derived for the associated expected warranty servicing cost per item sold. Second, using an example, for a given discretization of the warranty period, the way in which the number of subregions and their shape can be determined, so that the expected warranty servicing cost per item sold is minimum, is demonstrated. © 2007, Institution of Mechanical Engineers. All rights reserved.

An approach to modeling imperfect repairs under warranty settings is presented in Chukova, Arnold and Wang(12). They model the imperfect repairs using the concepts of delayed and accelerated distribution functions. As an extension of their approach, we design a procedure for estimating the degree of repair as well as other modeling parameters by Markov chain Monte Carlo (McMC) methods.

This paper compares the properties of various estimators for a beta-binomial model for estimating the size of a heterogeneous population. It is found that maximum likelihood and conditional maximum likelihood estimators perform well for a large population with a large capture proportion. The jackknife and the sample coverage estimators are biased for low capture probabilities. The performance of the martingale estimator is satisfactory, but it requires full capture histories. The Gibbs sampler and Metropolis-Hastings algorithm provide reasonable posterior estimates for informative priors.

Purpose - To provide a brief introduction to warranty analysis and a classification of general repairs. To introduce the notion of accelerated probability distribution and use it to model imperfect warranty repairs. Design/methodology/approach - The notion of accelerated probability distribution is discussed and its similarity with quasi-renewal and geometric processes is observed. An approach to modeling imperfect warranty repairs based on the accelerated probability distributions is presented, and the corresponding expected warranty cost over the warranty period under non-renewing free replacement warranty policy is evaluated. Findings - It is observed that quasi-renewal and the geometric processes are equivalent. Using data from an existing warranty database it is shown that the inter-repair times form a quasi-renewal process. The corresponding expected warranty cost over the warranty period under a non-renewing free replacement warranty policy is evaluated. Research limitations/implications - This approach is applicable only if the cost of the warranty repair is an increasing function of the number of repairs. Practical implications - Provides a useful approach to modeling inter-repair times incorporating the idea of imperfect repairs in practice. Originality/value - Provides an approach to model imperfect warranty repairs and to evaluate the corresponding expected warranty cost. © Emerald Group Publishing Limited.

The main focus of this study is on the modeling of the warranty claims and evaluating the warranty expenses. The cost of each warranty claim depends on the repair time associated with the claim. Alternating renewal process is used to model the operating and repair times. The warranty costs over the warranty period under renewing free replacement policy are evaluated. Also, the expected warranty expenses over the life cycle of the product are studied. Numerical examples illustrate the ideas.

A brief introduction to concepts and problems in warranty analysis is presented. The degree of warranty repair over the warranty period have an impact on the expected warranty costs and influences consumers' expenses over the post-warranty usage period of the product. Some techniques and approaches for modeling imperfect repairs are reviewed. A particular model is used to illustrate the impact of the degree of repair on warranty servicing costs.

The main focus of this study is on the modelling of the warranty claims and evaluating the warranty expenses. The cost of each warranty claim depends on the non-zero length of the repair time. Alternating renewal process is employed to model the operating and repair times. New results for alternating renewal process in finite horizon are derived. They are used to evaluate the warranty costs over the warranty period under non-renewing free replacement policy and over the life cycle of the product. Examples are provided to illustrate the ideas. Copyright (C) 2004 John Wiley Sons, Ltd.

A stochastic epidemic model with several kinds of susceptible is used to analyse temporal disease outbreak data from a Bayesian perspective. Prior distributions are used to model uncertainty in the actual numbers of susceptibles initially present. The posterior distribution of the parameters of the model is explored via Markov chain Monte Carlo methods. The methods are illustrated using two datasets, and the results are compared where possible to results obtained by previous analyses.

A frailty model for failure data is proposed to estimate the total number of faults in a system. The Littlewood model and Jelinski-Moranda are the two particular cases of the proposed formulation. The two-stage estimating procedure, a conditional likelihood and a Horvitz-Thompson estimator, is found to be efficient. Simulation studies are given to assess the performance of the estimator. Two examples axe also presented.

A new recapture debugging model is suggested to estimate the number of faults in a system, nu, and the failure intensity of each fault, phi. The Gibbs sampler and the Metropolis algorithm are used in this inference procedure. A numerical illustration suggests a notable improvement on the estimation of nu and phi compared with that of a removal debugging model.

We introduce one generalization of the mixed Poisson process referred to as the mixed Poisson-type process. An approach taken here is to assume the l(1)-isotropy of interevent times and to define the parameter as a function of observable quantities. An inhomogeneous variant of the new process is studied as a software reliability model. As an illustration a numerical example is analyzed via the Gibbs sampler. The mixed Poisson-type process is constructed through probabilistic behaviour of observable quantities and includes the mixed Poisson processes the limiting case. (C) 2000 Elsevier Science Ltd. All rights reserved.

In the literature on mixed Poisson processes, a number of characterisation properties have been studied. As a new characterisation property for mixed Poisson processes, we show that normalised event occurrence times are the order statistics of independent uniform random variables on (0, 1). Berman's theorem on l(p)-isotropic sequences is applied to prove the results.

Estimating the number of faults in a computer program is important in software debugging. A martingale equation is used to estimate the number of faults in removal-debugging by assuming a known proportionality constant between the failure rate of a 'newly detected fault' and a 'seeded fault'. The sensitivity of the assumption is examined, and the results are generalized to allow an unknown proportionality. The information of the proportionality is shown to be crucial in the precision & availability of the estimates. It is advisable to obtain the information about the proportionality constant from external sources in order to improve the efficiency of the method in this paper.

We introduce a new notion of positive dependence of survival times of system components using the multivariate arrangement increasing property. Following the spirit of Barlow and Mendel (J. Amer. Statist. Assoc. 87, 1116-1122), who introduced a new univariate aging notion relative to exchangeable populations of components, we characterize a multivariate positive dependence with respect to exchangeable multicomponent systems. Closure properties of such a class of distributions under some reliability operations are discussed. For an infinite population of systems our definition of multivariate positive dependence can be considered in the frequentist's paradigm as multivariate totally positive of order 2 with an independence condition. de Finetti(-type) representations for a particular class of survival functions are also given. (C) 1998 Elsevier Science B.V. All rights reserved.

We use an economic approach of Mendel to derive new bivariate exponential lifetime distributions. Features distinguishing this approach from the existing ones are (1) it makes use of the principle of indifference; (2) our parameter of interest is a measurable function of observable quantities; (3) the assessment of the probability measure for random lifetimes is performed by assessing that for random lifetime costs with a change of variables; and (4) characterization properties other than the bivariate loss-of-memory property are used to construct distributions. For the infinite population case, our distributions correspond to mixtures of existing bivariate exponential distributions such as the Freund distribution, the Marshall-Olkin distribution, and the Friday-Patil distribution. Moreover, a family of natural conjugate priors for Bayesian Freund(-type) bivariate exponential distributions is discussed.

A summaryof research outcomes: Research team of Waseda University Grantfor Special Research Projects (2014B-443) consisted of the following members: Principal researcher: Dr Yu Hayakawa,Waseda UniversityResearch collaborator: Dr Richard Arnold,Victoria University of WellingtonResearch collaborator: Dr Stefanka Chukova,Victoria University of WellingtonResearch collaborator: Ms Ting Ying Chen,Waseda University (undergraduate and masters student) In our research plan submitted for thisgrant, we stated that the main focus is on the modeling of the warranty costsbased on the generalised alternating renewal (GAR) process of type 1 (renewaloperating times followed by geometric repair times) and the GAR of type 2(geometric operating times followed by geometric repair times) under both thenon-renewing and renewing free replacement policies. We also obtain numericalresults by employing simulation methods.  We report that we have made good progresson this project and on some related projects as well. Dr Richard Arnold attended the 6th Asia-Pacific InternationalSymposium on Advanced Reliability and Maintenance Modelling held in Sapporoand visited the School of International Liberal Studies (SILS), WasedaUniversity in August 2014.  Yu Hayakawavisited Victoria University of Wellington from 23 February 2015 to 11 March2015, with which both Drs Richard Arnold and Stefanka Chukova are affiliated.  Throughout the period of this grant, we workedtogether on the project described above and a few other related topicsdescribed below.  We also worked on theproject with Dr Sarah Marshall, Auckland University of Technology, who joinedthe team in December 2014. 1)   We model the warranty claimsprocess and evaluate the warranty servicing costs under non-renewing andrenewing free replacement warranties.  Westudy two models which are based on the generalised alternating renewal processof type 1 (renewal operating times followed by geometric repair times – GAR I)and that of type 2 (geometric operating times followed by geometric repairtimes – GAR II) under both the non-renewing and renewing free replacementpolicies. We derive new results on GAR I & II with a finite time horizon.   Some results are in the manuscripts [1] & [5].  The manuscript [1] was presented by YuHayakawa at the 6th Asia-Pacific International Symposium on AdvancedReliability and Maintenance Modelling (APARM 2014, August 2014, Sapporo, Japan).  The manuscript [5] will be presented by YuHayakawa at The Ninth InternationalConference on Mathematical Methods in Reliability (MMR 2015) in Tokyo on 1-4June 2015. Dr Stefanka Chukova delivered an invited talk on this project atthe  International Statistical InstituteRegional Statistics Conference 2014 (ISI-RSC 2014) held in Kuala Lumpur,Malaysia, on 16-20 November 2014.  Thetalk title was "Warranty cost analysis: geometric operational and repairtimes." Dr Sarah Marshall will present some results on this project at the 27thEuropean Conference on Operational Research held in Glasgow, United Kingdom, 12-15thJuly 2015.  The talk title will be “Modellingwarranty costs using geometric repair times.” Ms Ting Ying Chen created some graphs to summarise simulationresults for this project. We are working on extended versions of the manuscripts [1] & [5]by incorporating the simulation results under both non-renewing and renewingwarranty replacement policies using GAR I and GAR II.  Also we are working on a numerical procedurerelated to some results included in the manuscript [1].   2)   We have constructed a model forcorrelated failures in multicomponent systems where failed components arerepaired.  Our results are in themanuscript [2].  This manuscript waspresented by Dr Richard Arnold at the 6th Asia-Pacific InternationalSymposium on Advanced Reliability and Maintenance Modelling (APARM 2014) inAugust 2014.  An expanded version of thepaper has been accepted for publication in the Journal of Risk and Reliability[4]. 3)   We apply finite mixture methodsto group units into fuzzy clusters, in order to model the joint distribution offailure times and severities.  This workis in the manuscript [3].  Thismanuscript was presented by Dr Richard Arnold at the 6thAsia-Pacific International Symposium on Advanced Reliability and MaintenanceModelling (APARM 2014) in August 2014. 4)   We develop inference proceduresfor multicomponent systems where the system is viewed as being made up ofindependent overlapping subsystems that we had previously published. Weintroduce a new type of grouping of subunits which allow for common causefailures, i.e., all the components within a subunit failuresimultaneously.  We extend our resultsand a paper on this project [6] is close to completion.  

A summary of research outcomes:Research team of Waseda University Grant for Special Research Projects (2013B-246) consisted of the following members:Principal researcher: Dr Yu Hayakawa, Waseda UniversityResearch collaborator: Dr Stefanka Chukova, Victoria University of WellingtonResearch collaborator: Yimo Xu, Waseda University (undergraduate student)Research assistant: Yan Liu, Waseda University (PhD student)In our research plan submitted for this grant, we stated that the main focus is on the modeling of the warranty claims and evaluation of the warranty servicing costs under non-renewing free replacement warranty policy. Our model is based the generalised alternating renewal process and we derive new results on this stochastic process with a finite time horizon. We also employ simulation methods to obtain numerical results. We report that we have made good progress on this project and on some related projects as well.Dr Stefanka Chukova visited the School of International Liberal Studies (SILS), Waseda University, during 12-19 December 2013. Dr Richard Arnold, her colleague from Victoria University of Wellington, was also visiting SILS on sabbatical from November 2013 to February 2014. Throughout the period of this grant, we worked together on the project described above and a few other related topics described below.1) We model the warranty claims process and evaluate the warranty servicing costs under non-renewing and renewing free replacement warranties. We use an increasing geometric process to model the consecutive repair times. This model is based on the generalised alternating renewal process (GAR) and we derive new results on GAR process with a finite time horizon. This work is in the manuscript [1] whose details are given in the next section. This manuscript will be presented at the 6th Asia-Pacific International Symposium on Advanced Reliability and Maintenance Modelling (APARM 2014, August 2014, Sapporo, Japan).Yan Liu conducted a simulation study on the above model under non-renewing replacement warranty and obtained numerical results. Yimo Xu wrote a set of notes on the relevant technical details and proofs of the theorems. We are working on an extended version of the manuscript [1] by incorporating the simulation results under both non-renewing and renewing warranty replacement policies (the manuscript [5]).2) We have constructed a model for correlated failures in multicomponent systems where failed components are repaired. Due to these repairs multiple failures of each component are observable. We assume that repair is instantaneous, but imperfect. Correlation amongst failures and the overall ageing of the system are modeled by treating each component failure as a shock which damages the other components in the system. Our results are in the manuscript [2]. This manuscript will be presented at the 6th Asia-Pacific International Symposium on Advanced Reliability and Maintenance Modelling (APARM 2014).3) Reliability data may incorporate a failure severity measure as well as the time of each failure. Such severities may however be qualitative ordinal data. We apply finite mixture methods to group units into fuzzy clusters, in order to model the joint distribution of failure times and severities. This work is in the manuscript [3]. This manuscript will be presented at the 6th Asia-Pacific International Symposium on Advanced Reliability and Maintenance Modelling (APARM 2014).4) We develop inference procedures for the shock models that we had previously published. We introduce a new type of grouping of subunits which allow for common cause failures, i.e., all the components within a subunit failure simultaneously. A paper on this project is in preparation.